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## G = C12×F5order 240 = 24·3·5

### Direct product of C12 and F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C12×F5
 Chief series C1 — C5 — C10 — D10 — C6×D5 — C6×F5 — C12×F5
 Lower central C5 — C12×F5
 Upper central C1 — C12

Generators and relations for C12×F5
G = < a,b,c | a12=b5=c4=1, ab=ba, ac=ca, cbc-1=b3 >

Subgroups: 156 in 60 conjugacy classes, 36 normal (20 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C2×C4, D5, C10, C12, C12, C2×C6, C15, C42, Dic5, C20, F5, D10, C2×C12, C3×D5, C30, C4×D5, C2×F5, C4×C12, C3×Dic5, C60, C3×F5, C6×D5, C4×F5, D5×C12, C6×F5, C12×F5
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C12, C2×C6, C42, F5, C2×C12, C2×F5, C4×C12, C3×F5, C4×F5, C6×F5, C12×F5

Smallest permutation representation of C12×F5
On 60 points
Generators in S60
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60)
(1 17 54 33 43)(2 18 55 34 44)(3 19 56 35 45)(4 20 57 36 46)(5 21 58 25 47)(6 22 59 26 48)(7 23 60 27 37)(8 24 49 28 38)(9 13 50 29 39)(10 14 51 30 40)(11 15 52 31 41)(12 16 53 32 42)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 56 39 35)(14 57 40 36)(15 58 41 25)(16 59 42 26)(17 60 43 27)(18 49 44 28)(19 50 45 29)(20 51 46 30)(21 52 47 31)(22 53 48 32)(23 54 37 33)(24 55 38 34)

G:=sub<Sym(60)| (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60), (1,17,54,33,43)(2,18,55,34,44)(3,19,56,35,45)(4,20,57,36,46)(5,21,58,25,47)(6,22,59,26,48)(7,23,60,27,37)(8,24,49,28,38)(9,13,50,29,39)(10,14,51,30,40)(11,15,52,31,41)(12,16,53,32,42), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,56,39,35)(14,57,40,36)(15,58,41,25)(16,59,42,26)(17,60,43,27)(18,49,44,28)(19,50,45,29)(20,51,46,30)(21,52,47,31)(22,53,48,32)(23,54,37,33)(24,55,38,34)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60), (1,17,54,33,43)(2,18,55,34,44)(3,19,56,35,45)(4,20,57,36,46)(5,21,58,25,47)(6,22,59,26,48)(7,23,60,27,37)(8,24,49,28,38)(9,13,50,29,39)(10,14,51,30,40)(11,15,52,31,41)(12,16,53,32,42), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,56,39,35)(14,57,40,36)(15,58,41,25)(16,59,42,26)(17,60,43,27)(18,49,44,28)(19,50,45,29)(20,51,46,30)(21,52,47,31)(22,53,48,32)(23,54,37,33)(24,55,38,34) );

G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60)], [(1,17,54,33,43),(2,18,55,34,44),(3,19,56,35,45),(4,20,57,36,46),(5,21,58,25,47),(6,22,59,26,48),(7,23,60,27,37),(8,24,49,28,38),(9,13,50,29,39),(10,14,51,30,40),(11,15,52,31,41),(12,16,53,32,42)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,56,39,35),(14,57,40,36),(15,58,41,25),(16,59,42,26),(17,60,43,27),(18,49,44,28),(19,50,45,29),(20,51,46,30),(21,52,47,31),(22,53,48,32),(23,54,37,33),(24,55,38,34)]])

C12×F5 is a maximal subgroup of   C30.3C42  D122F5  D605C4  (C4×S3)⋊F5

60 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C ··· 4L 5 6A 6B 6C 6D 6E 6F 10 12A 12B 12C 12D 12E ··· 12X 15A 15B 20A 20B 30A 30B 60A 60B 60C 60D order 1 2 2 2 3 3 4 4 4 ··· 4 5 6 6 6 6 6 6 10 12 12 12 12 12 ··· 12 15 15 20 20 30 30 60 60 60 60 size 1 1 5 5 1 1 1 1 5 ··· 5 4 1 1 5 5 5 5 4 1 1 1 1 5 ··· 5 4 4 4 4 4 4 4 4 4 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 4 4 4 4 4 4 type + + + + + image C1 C2 C2 C3 C4 C4 C4 C6 C6 C12 C12 C12 F5 C2×F5 C3×F5 C4×F5 C6×F5 C12×F5 kernel C12×F5 D5×C12 C6×F5 C4×F5 C3×Dic5 C60 C3×F5 C4×D5 C2×F5 Dic5 C20 F5 C12 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 2 8 2 4 4 4 16 1 1 2 2 2 4

Matrix representation of C12×F5 in GL5(𝔽61)

 50 0 0 0 0 0 13 0 0 0 0 0 13 0 0 0 0 0 13 0 0 0 0 0 13
,
 1 0 0 0 0 0 0 0 0 60 0 1 0 0 60 0 0 1 0 60 0 0 0 1 60
,
 11 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0

G:=sub<GL(5,GF(61))| [50,0,0,0,0,0,13,0,0,0,0,0,13,0,0,0,0,0,13,0,0,0,0,0,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,60,60,60,60],[11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,1,0] >;

C12×F5 in GAP, Magma, Sage, TeX

C_{12}\times F_5
% in TeX

G:=Group("C12xF5");
// GroupNames label

G:=SmallGroup(240,113);
// by ID

G=gap.SmallGroup(240,113);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-5,72,151,3461,599]);
// Polycyclic

G:=Group<a,b,c|a^12=b^5=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations

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